Nonlinear filters produce a nonparametric estimate of the probability density of state at each point in time. Currently-known nonlinear filters include Particle Filters and the Kushner equation (and its un-normalized version: the Zakai equation). However, these filters have limited measurement models: Particle Filters require measurement at discrete times, and the Kushner and Zakai equations only apply when the measurement can be represented as a function of the state. We present a new nonlinear filter for continuous-time measurements with a much more general stochastic measurement model. It integrates to Bayes' rule over short time intervals and provides Bayes-optimal estimates from quantized, intermittent, or ambiguous sensor measurements. The filter has a close link to Information Theory, and we show that the rate of change of entropy of the density estimate is equal to the mutual information between the measurement and the state and thus the maximum achievable. This is a fundamentally new class of filter that is widely applicable to nonlinear estimation for continuous-time control.